Deriving Schrödinger's Equation From My Fundamental Equation

[AnssiH]

Rade, I posted a reply to:

 

http://hypography.com/forums/philosophy-of-science/11733-what-can-we-know-of-reality-29.html#post264413

[Rade]

Rade, I posted a reply to:

 

http://hypography.com/forums/philosophy-of-science/11733-what-can-we-know-of-reality-29.html#post264413

 

AnssiH....THANK YOU....I will provide further comments-questions at this new link and not post on this thread anymore....and to your question....

..."I can only hope this post is helpful to you, and I hope you appreciate the effort"...

.....the answer is a very appreciated yes.

[AnssiH]

The term:

[math]\left\{ 2K^2\frac{\partial^2}{\partial t^2} + q^2 \right\}[/math]

 

(which is correct) does not posses that important negative sign and thus (as written) does not factor. That is why it was changed to

[math]\left\{ 2K^2\frac{\partial^2}{\partial t^2} -(i q)^2 \right\}[/math]

 

Ah, right.

 

My non-sophisticated mathematical mind thinks it would have been little bit clearer if the [imath]-(i q)^2[/imath] had not been factored to [imath]q^2[/imath] in the previous step, only to be turned back to [imath]-(i q)^2[/imath] in the next turn... :I But then, I guess it makes no difference to anyone who actually knows their math.

 

 

At this point, I will invoke a third approximation. I will concern myself only with cases where [imath]K\sqrt{2}\frac{\partial}{\partial t}\vec{\Phi} \approx -iq\vec{\Phi}[/imath] to a high degree of accuracy.

 

I don't understand what that approximation means conceptually, maybe you can elaborate on that. I.e., what is ultimately the justification for the approximation.

 

Since I don't understand what it means, I'll just take it on faith to get onwards:

 

In this case, the first term on the right may be replaced by -2iq and, after devision by 2q, we have

[math]\left\{\frac{1}{2q}\frac{\partial^2}{\partial x^2}+\frac{1}{2q}G(x)\right\}\vec{\Phi}(x,t)= -i\left\{\sqrt{2}K \frac{\partial}{\partial t} + iq \right\}\vec{\Phi}(x,t).[/math]

 

Okay, that looks all valid to me.

 

And I guess here you have just chosen to not substitute the [imath]\sqrt{2}K \frac{\partial}{\partial t}[/imath] with [imath]-iq[/imath] in the second term, even though (or because?) it would just remove the whole term... ?

 

Once again, the form of the equation suggests we redefine [imath]\vec{\Phi}[/imath] via an exponential adjustment [imath]\vec{\Phi}(x,t)=\vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}}[/imath], thus simplifying the differential equation by removing the final iq term.

 

Well, I wrote down:

[math]\left\{\frac{1}{2q}\frac{\partial^2}{\partial x^2}+\frac{1}{2q}G(x)\right\} \vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}} = -i\left\{\sqrt{2}K \frac{\partial}{\partial t} + iq \right\} \vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}}[/math]

 

And I've been staring at that for little while now but I just don't know which way to move with it... :I

 

Help!

 

-Anssi

[Doctordick]

My non-sophisticated mathematical mind thinks it would have been little bit clearer if the [imath]-(i q)^2[/imath] had not been factored to [imath]q^2[/imath] in the previous step, only to be turned back to [imath]-(i q)^2[/imath] in the next turn... :I But then, I guess it makes no difference to anyone who actually knows their math.

You are, of course, correct. Mathematicians often simplify things all the time without considering the possible confusion it might create. It never even dawned on me that I had done that. I should not have as it served no purpose.

I don't understand what that approximation means conceptually, maybe you can elaborate on that. I.e., what is ultimately the justification for the approximation.

The justification is a couple of lines later in that same post. I was shown that factorization by the same professor who taught me introductory quantum mechanics (the same one to whom I explained my childish solution to relativity way back in 1964). He showed it to me in terms of the Klein–Gordon equation.

[math]\frac{1}{c^2}\frac{\partial^2}{\partial t^2}\Psi -\nabla^2\Psi +\frac{m^2c^2}{\hbar^2}\Psi =0 [/math]

 

Which is normally referred to as the relativistic expression of Schrödinger's equation. By convention, all effects of the interactions is usually presumed to be embedded in “m”. If that is not “presumed”, and, instead, the last term is written

[math]\frac{m_0^2c^2}{\hbar^2}\Psi +V\Psi[/math]

 

where “[imath]m_0[/imath]” is taken to be “the rest mass”, the term

[math]\frac{1}{c^2}\frac{\partial^2}{\partial t^2} +\frac{m_0^2c^2}{\hbar^2} [/math]

 

can be factored in exactly the way I have just factored it, in which case, the Kline-Gordon equation (for non relativistic situations, becomes Schrödinger's equation with only a minor alteration in the definition of V: a shift by a constant). Years ago, (when I first used that factorization to deduce Schrödinger's equation from my equation) I went to tell him about it. I was told that he was suffering from Alzheimer's and not available to talk to anyone. Isn't that the way life is?

And I guess here you have just chosen to not substitute the [imath]\sqrt{2}K \frac{\partial}{\partial t}[/imath] with [imath]-iq[/imath] in the second term, even though (or because?) it would just remove the whole term... ?

The second term reduces to almost zero (it can't actually be zero because this is an approximation). The first term is roughly 2q which is not anywhere near zero so it can be considered very similar to 2q (only off by a small percentage). So, yes; it would unjustifiably remove the whole term and essentially remove the time derivative (which will be defined to be energy) from the equation.

Well, I wrote down:

[math]\left\{\frac{1}{2q}\frac{\partial^2}{\partial x^2}+\frac{1}{2q}G(x)\right\} \vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}} = -i\left\{\sqrt{2}K \frac{\partial}{\partial t} + iq \right\} \vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}}[/math]

 

And I've been staring at that for little while now but I just don't know which way to move with it... :I

Just work with the right hand side. First, multiply it out and get the following

[math]-i\left\{\sqrt{2}K \frac{\partial}{\partial t} + iq \right\} \vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}}= -i\sqrt{2}K\frac{\partial}{\partial t} \vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}}+q \vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}}[/math]

 

Now, for the moment, look at that first term,

[math]-i\sqrt{2}K\frac{\partial}{\partial t}\vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}}[/math].

 

That is a product of two different functions of t. The product rule of differentiation says that such a differential yields two terms which are: the differential of the first term times the second term plus the first term times the differential of the second term. Explicitly, that would be

[math] -i\sqrt{2}K\frac{\partial}{\partial t} \vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}}= -i\sqrt{2}K\left\{\frac{\partial}{\partial t} \vec{\phi}(x,t)\right\}e^{\frac{-iqt}{K\sqrt{2}}}-i\sqrt{2}K\vec{\phi}(x,t)\frac{\partial}{\partial t}e^{\frac{-iqt}{K\sqrt{2}}}=[/math]

 

[math]-i\sqrt{2}K\left\{\frac{\partial}{\partial t}\vec{\phi}(x,t)\right\}e^{\frac{-iqt}{K\sqrt{2}}} -i \sqrt{2}K\vec{\phi}(x,t)\frac{-iq }{K\sqrt{2}}e^{\frac{-iqt}{K\sqrt{2}}}=-i\sqrt{2}K\left\{\frac{\partial}{\partial t}\vec{\phi}(x,t)\right\}e^{\frac{-iqt}{K\sqrt{2}}} -q\vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}}[/math]

 

Now, go back and substitute that for the first term (when we multiplied the right hand side out). The right hand side will now be

[math]-i\sqrt{2}K\left\{\frac{\partial}{\partial t}\vec{\phi}(x,t)\right\}e^{\frac{-iqt}{K\sqrt{2}}} -q\vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}}+q \vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}}[/math]

 

And the last two terms exactly cancel out. Going back to the equation we started with (as it now appears, without those two last terms), we have

[math]\left\{\frac{1}{2q}\frac{\partial^2}{\partial x^2}+\frac{1}{2q}G(x)\right\} \vec{\phi}(x,t)e^{\frac{-iqt}{K\sqrt{2}}} = -i\sqrt{2}K \left\{\frac{\partial}{\partial t}\vec{\phi}(x,t)\right\}e^{\frac{-iqt}{K\sqrt{2}}}[/math]

 

where the exponential function explicitly appears in every term but is not to be differentiated anywhere and it may be factored out (we can divide it out or we can multiply the whole equation from the right by [imath]e^{\frac{-iqt}{K\sqrt{2}}}[/imath], it all results in exactly the same effect). We are left with the equation

[math]\left\{\frac{1}{2q}\frac{\partial^2}{\partial x^2}+\frac{1}{2q}G(x)\right\} \vec{\phi}(x,t) = -i\sqrt{2}K\frac{\partial}{\partial t}\vec{\phi}(x,t)[/math]

 

That mathematical maneuver is well known to anyone who does classical quantum mechanics. The right hand side of that equation is a conserved quantity (which will soon be identified with energy). The mathematical maneuver I just introduced you to simply changes that conserved quantity by a constant. In classical physics, the zero reference for energy can always be changed by a constant and has no physical consequences at all. In modern physics people tend to believe this characteristic of physics is no longer true and they come up with a thing they call “zero point energy” which I suspect is actually a figment of their imagination; of course, I tend to believe the fact that reality obeys physics is a figment of their imagination.

 

Only Anssi will understand the meaning of that comment so don't everybody else jump off the deep end!

 

Have fun -- Dick

[AnssiH]

Ah, okay, I was able to walk through all of that succesfully, after some head scratching and using google calculater to figure out what -i*-i...

 

I didn't stop and think about that commentary about the Klein-Gordon equation too deeplythough and don't understand it completely. I suppose it was just additional commentary here anyway.

 

...to be continued soon...

 

-Anssi

[AnssiH]

We are left with the equation

[math]\left\{\frac{1}{2q}\frac{\partial^2}{\partial x^2}+\frac{1}{2q}G(x)\right\} \vec{\phi}(x,t) = -i\sqrt{2}K\frac{\partial}{\partial t}\vec{\phi}(x,t)[/math]

 

To anyone familiar with modern physics, the equation should be beginning to look very familiar. In fact, if we multiply through by [imath]-\hbar c[/imath] (which clearly has utterly no impact on the solution as it multiplies every term)

 

My attempt to write that step down is:

 

[math]\left\{\frac{-\hbar c}{2q} \frac{\partial^2}{\partial x^2} + \frac{-\hbar c}{2q} G(x)\right\} \vec{\phi}(x,t) = -\hbar c (-i)\sqrt{2}K \frac{\partial}{\partial t}\vec{\phi}(x,t)[/math]

 

I can only hope it is correct... :I

 

Assuming it is, I tried to follow the next step:

 

and make the following definitions directly related to constants already defined,

[math]m=\frac{q\hbar}{c}[/math] , [math]c=\frac{1}{K\sqrt{2}}[/math] and [math]V(x)= -\frac{\hbar c}{2q}G(x)[/math]

 

it turns out that the equation of interest (without the introduction of a single free parameter: please note that no parameters not defined in the derivation of the equation have been introduced) is exactly one of the most fundamental equations of modern physics.

[math]\left\{-\left(\frac{\hbar^2}{2m}\right)\frac{\partial^2}{\partial x^2}+ V(x)\right\}\vec{\phi}(x,t)=i\hbar\frac{\partial}{\partial t}\vec{\phi}(x,t)[/math]

 

Well, I figure the first definition [imath]m=\frac{q\hbar}{c}[/imath] (mass?) comes to play in the first term, and after a lot of head scratching, I was finally able to figure out the steps, when I did it in reverse. At least I hope everything I do here is valid (just had to trust my logic but I know it's easy for me to make errors):

 

So I started with your end result, and substituted the [imath]m[/imath] with [imath]\frac{q\hbar}{c}[/imath]

 

[math]-\left(\frac{\hbar^2}{2m}\right) = -\left( \frac{\hbar^2} {2 \left( \frac{q\hbar}{c} \right)} \right)[/math]

 

I suppose that can be written as:

 

[math]-\left(\frac{\hbar^2}{2} \cdot \frac{c}{q\hbar} \right)[/math]

 

Which seems to factor to what I had:

[math]\frac{-\hbar c}{2q}[/math]

 

So if that's all valid, I think understand the left hand side of...

 

[math]\left\{-\left(\frac{\hbar^2}{2m}\right)\frac{\partial^2}{\partial x^2}+ V(x)\right\}\vec{\phi}(x,t)=i\hbar\frac{\partial}{\partial t}\vec{\phi}(x,t)[/math]

 

...since the second term, [imath]V(x)[/imath], seems to be just a straightforward substitution (except I don't know what it represents, the wiki page for Schrödinger equation talks about "time independent potential energy of particle", but I don't understand what that means either)

 

Finally, I'm focusing on the right hand side, substituting [math]c[/math] with [math]\frac{1}{K\sqrt{2}}[/math]:

 

[math] -\hbar c \left(-i\sqrt{2}K\right) \frac{\partial}{\partial t}\vec{\phi}(x,t) = -\hbar \frac{1}{K\sqrt{2}} (-i)\sqrt{2}K \frac{\partial}{\partial t}\vec{\phi}(x,t)

[/math]

 

I suppose [imath]\sqrt{2}K[/imath] factors right out, leaving me with:

 

[math] -\hbar (-i) \frac{\partial}{\partial t}\vec{\phi}(x,t)

[/math]

 

Since your end result is:

 

[math]i\hbar\frac{\partial}{\partial t}\vec{\phi}(x,t)[/math]

 

I suppose the negatives just factor out normally, even with the imaginary number involved. If that's a correct assumption, I think I understand this part too...

 

All looks valid?

 

-Anssi

[Doctordick]

Hi Anssi, I am sorry it took me so long to respond to your post. I left it to last because I wanted to give you serious attention: i.e., I wanted to get the trivial things (the other posts) out of the way first but they just kept posting and I couldn't get finished. I finally gave up on that agenda and went to your post.

...since the second term, [imath]V(x)[/imath], seems to be just a straightforward substitution (except I don't know what it represents, the wiki page for Schrödinger equation talks about "time independent potential energy of particle", but I don't understand what that means either)

Sorry I made the thing so difficult for you. Instead of writing the first term in the form m=, I should divided both sides by qm and obtained the expression 1/q = and then it also would have been a simple substitution.

[math]\frac{1}{q}= \frac{\hbar}{mc} [/math]

 

But you figured it out anyway. You did fine.

All looks valid?

Yes, everything you did was valid (a little round about but absolutely valid).

 

Regarding the second term, as per wiki, V(x) is the time independent potential energy of the particle (per Schrödinger). In my equation it is the the change in the energy of the universe due the change in position of the element of interest. Remember, we removed that component of the energy of the rest of the universe which didn't change (that S_r term) a long time ago. So this is an algebraic function which tells you how the potential energy changes with respect to position of the element of interest. Mine is a consequence of the direct calculation of energy of the whole universe while Schrödinger's is a phenomenological function determined by experiment. They both refer to exactly the same phenomena.

 

Thus my equation is exactly Schrödinger's equation without introducing any constants of any kind. You should be aware of the fact that K establishes a fixed velocity which I call v_? in that thread An “analytical-metaphysical” take on Special Relativity! You need to understand that thread in order to understand where c comes from (that is, the fact that it is not really a free parameter).

 

Before we go on to Dirac's equation, we probably ought to go through the Special Relativity thread first. Other than that, I think we are ready to go on.

 

Have fun -- Dick

[AnssiH]

Yes, everything you did was valid (a little round about but absolutely valid).

 

Heh, so I suspected.... Great.

 

Regarding the second term, as per wiki, V(x) is the time independent potential energy of the particle (per Schrödinger). In my equation it is the the change in the energy of the universe due the change in position of the element of interest. Remember, we removed that component of the energy of the rest of the universe which didn't change (that S_r term) a long time ago. So this is an algebraic function which tells you how the potential energy changes with respect to position of the element of interest. Mine is a consequence of the direct calculation of energy of the whole universe while Schrödinger's is a phenomenological function determined by experiment. They both refer to exactly the same phenomena.

 

Well, my understanding of Schrödinger's Equation is quite shallow, (I don't know how it is conventionally looked at) and I would like to understand it a bit better... Don't really know what potential energy means in quantum mechanical context :I

 

Thus my equation is exactly Schrödinger's equation without introducing any constants of any kind. You should be aware of the fact that K establishes a fixed velocity which I call v_? in that thread An “analytical-metaphysical” take on Special Relativity! You need to understand that thread in order to understand where c comes from (that is, the fact that it is not really a free parameter).

 

Before we go on to Dirac's equation, we probably ought to go through the Special Relativity thread first. Other than that, I think we are ready to go on.

 

Well, I read the rest of the OP, and I can understand what you are saying there little bit. Not everything, but I understand that you are drawing out the relationship [imath]E = mc^2[/imath] - at this point seems completely plausible to me that that relationship was embedded to the original constraints. And you defined Energy, Momentum and Mass as conserved quantities that, I imagine, correspond very much to how they are defined in modern physics.

 

If that's enough then yes, let's concentrate on the special relativity for a while...

 

-Anssi

[Doctordick]

Well, my understanding of Schrödinger's Equation is quite shallow, (I don't know how it is conventionally looked at) and I would like to understand it a bit better... Don't really know what potential energy means in quantum mechanical context :I

Fundamentally, it is little more than a statement of energy conservation: kinetic energy (the energy of motion) plus potential energy (the energy of position) equals total energy. The kinetic energy is related to the momentum, a spacial differential and the total energy is related to the time derivative. These differential relationships actually arise through mathematical analysis of procedures designed to solve Newton's equations. They also yield our understanding of electromagnetic phenomena as expressed by Maxwell's equations: they connect the momentum and energy of a photon to Newton's idea of momentum and energy. I wouldn't really worry about it if I were you; just take it as the definition of energy and momentum (and, in my case, mass). It turns out that it fits exactly the phenomena Newton defined to be momentum, energy and mass (via connections to experiments: i.e., expectations given by [imath]\psi^\dagger(\vec{x},t)O\psi(\vec{x},t) dx[/imath] where O stands for an operator). Except for a little uncertainty. :lol:

If that's enough then yes, let's concentrate on the special relativity for a while...

Wonderful; I will consider this thread to be essentially closed.

 

See you on the other thread -- Dick

[Doctordick]

I will consider this thread to be essentially closed.

Perhaps I was a bit hasty there. I seem to be still referring to this thread quite often and it could use a sticky tag to keep things straight. :shrug:

 

Have fun -- Dick

[Doctordick]

I just spent the whole day fixing the opening post to this thread and find myself perturbed by the length of time that fix took. The new forum is unbelivably slow compared to what we had. Since this thread is very important to all my posts, I decided to add this entry to bring it back to the front page. Making this simple post took almost three minutes; most of the time spent waiting for the web site's information to download to my machine. The old site used to do that quicker than it took to notice.

 

Essentially, understanding the opening post to this thread is the very center of understanding what I am talking about; if anyone has any interest in understanding me, I suggest they digest the OP on this thread.

 

Have fun -- Dick

[AnssiH]

The math on the opening post is broken (which DD may or may not fix when he notices), but either way, a probably better representation of the argument can be read from his book, which is available at the http://foundationsofphysics.blogspot.com

 

I just posted a blog post related directly to the OP of this thread, so I decided to post a copy here also;

 

About the meaning of Schrödinger equation

 

It can be little bit confusing - especially through the first chapters - to grasp what exactly is being developed in the analysis (see the book on the right). It is easy to miss that what is being developed is not a physical theory, but about physical theories; every argument exists an abstraction level above physics. Especially the first chapters in themselves are highly abstract as they are meticulously detailing an epistemological framework for an analysis (a framework that is most likely completely unfamiliar to the reader).

 

I find that typically people start to get a better perspective towards the analysis when Schrödinger equation suddenly appears in plain sight in rather surprising manner.

 

So to provide a softer landing, I decided to write some commentary and thoughts on that issue. Don't mistake this for an analytical defense of the argument; for that it's better to follow the arguments in the book. I'm merely providing a quick overview, and pointing out some things worth thinking about.

 

Let me first jump forward to page 51, equation (3.15) is;

[math]

-\frac{\hbar^2}{2m}\frac{\partial^2}{\partial x^2}\Psi(x,t)+V(x)\Psi(x,t)=i\hbar\frac{\partial}{\partial t}\Psi(x,t)

[/math]

which is identical to Schrödinger equation in one dimension for a single particle.

 

In Richard's analysis this expression arises from radically different considerations than its historical twin, and the steps that were taken to get here offer some insight as to why Schrödinger's equation is a valid approximated general description of the entire universe.

 

To get some perspective on the issue, let me do a quick hand-wavy run through some of the history of Schrödinger equation.

 

Usually the [math]\Psi[/math] in Schrödinger equation is interpreted as representing a probability amplitude (whose absolute square is the probability of observation results). Erwin Schrödinger didn't arrive to this representation by trying to describe probability amplitude's per se. Rather it arose from a series of assumptions and definitions that were simply necessary to explain some observations, under certain pre-existing physics definitions;

 

Max Planck created a concept of "discrete energy bundles" as part of his explanation for "black body radiation". Motivated by this, Albert Einstein suggested that electromagnetic radiation exists in spatially localized bundles - call them "photons" - while they would still preserve their wave-like properties where suitable. Consequently, Louis de Broglie showed that wave behavior would also explain observations related to electrons, and further hypothesized that all massive particles could be associated with waves describing their behavior.

 

Schrödinger set out to find wave equation that would represent the behavior of an electron in 3 dimensions; in particular he set out to find a wave equation that would correctly reproduce the observed emission spectrum of a hydrogen atom.

 

Schrödinger arrived at a valid equation, which would "detail the behavior of wave function [math]\Psi[/math] but say nothing of its underlying nature". There is no underlying reasoning that would have directly led Schrödinger to this particular equation, there is just the simple fact that this is the form that yields valid probabilistic expectations for the observed measurement outcomes.

 

It was only after the fact that the desire to explain what in nature makes Schrödinger equation valid, would lead to a multitude of hypothetical answers (i.e. all the ontological QM interpretations).

 

Effectively what we know is that the equation does correctly represent the probabilities of measurement outcomes, but all the ideas as to why, are merely additional beliefs.

 

Just as a side-note, every step on the historical route towards Schrödinger equation represents a small change in the pre-existing physics framework; every step contains an assumption that the pre-existing framework represents reality mostly correctly. Physical theories don't just spring out form radically new perspectives, but rather they tend to be sociologically constructed as logical evolution from previous paradigms, generalized to explain a wider range of phenomena. These successful generalizations may cause revolutionary paradigm shifts, as was the case with Schrödinger equation.

 

Alright, back to Richard's analysis. I will step backwards from the equation (3.15) (Schrödinger Equation) to provide a quick but hand-wavy preview of what is it all about.

 

First note that [math]\Psi[/math]is indeed here also related to the probability of a measurement outcome via [math]P = \Psi^{\dagger} \cdot \Psi[/math]. But it has not been interpreted as so after the fact; rather it has been explicitly defined at the get-go as any unknown function that yields observed probabilities in self-consistent fashion; any function that does so can be seen as a valid explanation to some data.

 

For more detail on this definition, see equation (2.13) on page 33 and the arguments leading up to it. Note especially that the definition (2.13) is specifically designed to not exclude any possible functions embedded inside [math]\Psi[/math]. It is important that this move does not close out any possibilities pre-emptively; if it did, we would have just made an undefendable assumption about the universe. The definition (2.13) will have an impact on the mathematical appearance of any expressions, but this is most correctly seen as an abstract (and in many ways arbitrary) mathematical terminology choice. Its consequences should be viewed as purely epistemological (purely related to the mental terminology embedded to our explanation), not ontological (relating to nature in itself). E.g. the squaring comes from the definition of probability, and [math]\Psi[/math]being complex function simply impacts the apparent form of its constraints (which play important role in the form of Schrödinger equation, as becomes evident little later).

 

Let's jump to equation (3.14). which is logically equivalent to Schrödinger equation; only couple of simple algebraic steps stand between the two expressions. There is absolutely no reason to do these steps other than to point out that the equations indeed do represent the same constraint.

 

As is mentioned in the book, (3.14) was obtained as a starting point for a perturbation attack, to find the constraints on a [math]\Psi[/math]for a single element, in an unknown universe (under few of conditions, which I'll return to).

 

To get some idea of what that means, let me first point out that the underlying constraints embedded into the equation have a deceptively simple source. Equations tagged as (2.7) on page 26 are;

 

[math]

\sum_{i=1}^{n}\frac{\partial}{\partial \tau_i}P(x_1, \tau_1, x_2, \tau_2 ... , x_n, \tau_n,t)=0

[/math]

and

 

[math]

\sum_{i=1}^{n}\frac{\partial}{\partial x_i}P(x_1, \tau_1, x_2, \tau_2 ... , x_n, \tau_n,t)=0

[/math]

which simply means that, under any explanation of any kind of universe, the probability of an outcome of any measurement is always a function of some data points that provide a context (the meaning) for that measurement. But the probability is never a function of the assignment of labels (names) to those data points.

 

Alternatively you can interpret this statement in terms of an abstract coordinate system [math](x, \tau)[/math](a view also developed carefully in the book), in which case we could say, the probability of an outcome of a measurement is not a function of the location of the context inside the coordinate system. Effectively that is to say that the defined coordinate system does not carry any meaning with its locations. After all, it is explicitly established as a general notation capable of representing any kind of explanation.

 

Note that what the data points are, and what they mean, is a function of each and every possible explanation. Thus the only constraints that are meaningful to this analysis are those that would apply to any kind of assignment of labels.

 

Note that exactly similar symmetry constraint is defined for partial derivative of [math]t[/math], the index for time-wise evolution. See (2.19) where it is expressed against [math]\Psi[/math]

 

The other type of underlying constraint is represented in equation (2.10) with a Dirac delta function, meaning at its core that different data points cannot be represented as the same data point by any self-consistent explanation; a rather simple epistemological fact.

 

The definition of [math]\Psi[/math]as [math]P = \Psi^{\dagger} \cdot \Psi[/math]and some algebra will lead to a succinct equation expressing these universal epistemological constraints as single equation (2.23)

 

[math]

\left \{ \sum_i \vec{\alpha}_i \cdot \vec{\triangledown}_i + \sum_{i \neq j} \beta_{ij} \delta(\vec{x}_i - \vec{x}_j) \right \}\Psi=\frac{\partial}{\partial t}\Psi = im\Psi

[/math]

which was somewhat useless for me to write down here as you need to view the associated definitions in the book anyway to understand what it means. You can see page 39 for definitions of the alpha and beta elements, and investigate the details of this expression better there. For those who want to just carry on for now, effectively this amounts to be a single expression representing exactly the above-mentioned constraints - without creating any additional constraints - on [math]\Psi[/math].

 

View this as a constraint that arises from the fact that any explanation of anything must establish a self-consistent terminology to refer to what-is-being-explained, and this is the constraint that any self-consistent terminology in itself will obey, regardless of what the underlying data is.

 

Chapter 3 describes the steps from this point onward, leading us straight into Schrödinger's expression. It is worth thinking about what those steps actually are.

 

First steps are concerned with algebraic manipulations to separate the collection of elements into multiple sets under common probability relationship P(set #1 and set #2) = P(set #1)P(set #2 given set #1) (page 45)

 

Leading us to equation (3.6), which is an exact constraint that a single element must obey in order to satisfy the underlying epistemological constraints. But this expression is still wonderfully useless since we don't know anything about the impact of the rest of the universe (the [math]\Psi_r[/math])

 

From this point on, the moves that are made are approximations that cannot be defended from an ontological point of view, but their epistemological impact is philosophically significant.

 

The first move (on page 48) is the assumption that there is only negligible feedback between the rest of the universe and the element of interest. Effectively the universe is taken as stationary in time, and the element of interest is assumed to not have an impact to the rest of the universe.

 

Philosophically this can be seen in multiple ways. Personally I find it interesting to think about the fact that, if there exists a logical mechanism to create object definitions in a way where those objects have negligible feedback to the rest of the universe, then there are rather obvious benefits in simplicity for adopting exactly such object definitions, whether or not those definitions are real or merely a mental abstraction.

 

Note further that if it was not possible - via reasonable approximations or otherwise - to define microscopic and macroscopic "objects" independently from the rest of the universe, so that those objects can be seen as universes unto themselves, the alternative would be that any proposed theory would have to constantly represent state of the entire universe. I.e. the variables of the representation would have to include all represented coordinates of everything in the universe simultaneously.

 

That is to say, whether or not reality was composed of complex feedback loops, any method of modeling probabilistic expectations with as little feedback mechanisms as possible would be desirable, and from our point of view such explanations would appear to be the simplest way to understand reality.

 

Next steps are just algebraic moves under the above assumption, leading to equation (3.12) on page 50. Following that equation, the third and final approximation is set as;

 

[math]

\frac{\partial}{\partial t} \Psi \approx -iq\Psi

[/math]

which leads to an expression that is already effectively equivalent to Schrödinger's equation, simply implying that this approximation plays a role in the exact form of Schrödinger's Equation. See more commentary about this from page 53 onward.

 

And there it is, the equation that implies wave particle duality to the entire universe, and yielded a revolution in the concepts of modern physics, arises from entirely epistemological constraints, and few assumptions that are forced upon us to remove overwhelming complexity from a representation of a universe.

 

The steps that got us here tell us exactly what makes Schrödinger Equation generally valid. When we create our world view, we define the elements (the mental terminology) with exactly the same epistemological constraints that would also yield Schrödinger Equation in Richard's analysis. The only difference between different representations (everyday, classical, or quantum mechanical) is that different approximations are made for simplicity's sake.

 

The steps that the field of physics took towards Schrödinger equation were always dependent on the elements that had already been defined as useful representation of reality. They were merely concerned of creating generalized expressions that would represent the behavior of those elements.

 

The so-called quantum mystery arises from additional beliefs about the nature of reality - redundant beliefs that the elements we define as part of our understanding of reality, are also ontological elements in themselves. There exists many different beliefs (QM interpretations) that each yield a possible answer to the nature behind quantum mechanics, but scientifically speaking, there is no longer need to explain the validity of Schrödinger Equation from any hypothetical ontological perspective.

 

So it appears the critical mistake is to assume that the particles we have defined are also in themselves real objects, from which our understanding of reality arises. Rather the converse appears to be true; a useful method of representing the propagation of probabilistic expectations between observations is driving what we have define as objects, and consequently this epistemological issue critically affects how do we view the universe meaningfully in the first place. After all, understanding reality is meaningful only in so far that we can correctly predict the future, and the only meaningful object definitions have to be directly related to that fact.

 

Thus, to avoid redundant beliefs in the subject matter, Schrödinger equation can be seen simply as a valid general representation of the propagation of our expectations (of finding a defined particle) between observations, governed by the same constraints that govern what we define as "objects". The exact reason why the particles "out there" appear to be reacting to our mere observations is that the pure observation X implies the existence of a particle Y only in our own mental classification of reality. That is why the particles that we have defined do not behave as particles would in-between observations.

 

To assume the ontological existence of associated particles as we have defined them, is not only redundant, but also introduces a rather unsolvable quantum mystery.

Edited July 16, 2014 by AnssiH

[Rade]

Anssih,

 

Thank you for the above post, as I find it very helpful.

 

I completely disagree with this comment you made:

 

"So it appears the critical mistake is to assume that the particles we have defined are also in themselves real objects"

 

First, one does not assume that any particles we define are in-themselves real objects, one knows they are real objects via the axiomatic fact that they exist. Next one also knows logically that the in-themselves nature of these defined particles is unknown and unknowable unless the knowledge comes from inside the particle (this is what being in-themselves means).

 

There is absolutely no 'critical mistake' to know that particles that exist as real objects that are experienced by a human with unknown in-themselves nature have the potential to be differentiated from other such particles and thus formed into mental CONCEPTS. There is no critical mistake to then assign context dependent DEFINITIONS on these concepts to allow humans to communicate how such real particles are different from each other, and perhaps of greater value, to communicate among humans how such real particles can be mentally integrated into new concepts that allow for new definitions.

 

My take on what DD has done, which I think is important and very interesting, is that he derives equations of physics such as the one by Schrödinger via DEFINITIONS that humans place on CONCEPTS of real objects, rather than the more direct approach of deriving the equations via observations on the CONCEPTS of the real objects. Schrödinger derived his equation via CONCEPTS of real objects and thus it is valid for all contextually valid DEFINITIONS of the concepts. DD has apparently (the case is still open) discovered how and why all such valid definitions of valid physical concepts give approximately the same equation as Schrödinger, but this does not mean that we no longer need the Schrödinger approach. In fact, the approach of DD is meaningless until a genius such as Schrödinger or Dirac or Einstein first puts forth the valid conceptual equation derived from real phenomenon of real objects, prior to any attempt to place concept or definition place holders on them.

 

Hats off to DD if his definitional approach to derive conceptual equations of physics makes it through critical review by his peers (physicists), which means either he (or you as his proxi) will have to submit the most recent edition of his 'book' to the PHYSICS FORUM section of this Hypography Science website, where the 2007 version of his "Foundations of Physical Reality' already has been critically reviewed and for the most part rejected by professional physicists.

[AnssiH]

Hey Rade.

 

You know I find this Ayn Rand terminology to be completely useless when it comes to describing the problem that Kant was describing. And it's almost amusing how objectivist epistemology and Kantian philosophies are miscommunicating because they are re-arranging terminologies in the way that the same words are used to refer to different things;

 

Interpret Randian communication from Kantian terminology, and it comes out as thoughtless naive realism written by a 6 year old, which is not at all how it was intented.

 

Interpret Kantian communication from Randian terminology, and it comes out as purely useless idealistic assumption, also not at all how it was intented.

 

Obviously I can't know with certainty what Rand (or Kant) was thinking but her comments about Kantian philosophy suggest that she interpreted Kant as suggesting our mental faculties are to be considered somehow faulty.

 

It's like viewing the "map is not the territory" problem as suggesting that the world is imaginary. Or viewing the statement "photon does not exist" as a statement that we imagine everything.

 

So where a Kantian philosophy is saying "Photon does not exist" (emphasis on definition of photon), she hears it as "Photon does not exist" (emphasis on the existence of anything)

 

Her emphasis on defining "existence" as an axiom in itself (and axiom as tautology of sort) appears to be a result of her thinking that Kantian philosophy is trying to solve a problem that is not a problem.

 

Her emphasis on "existence" implies she is addressing an issue that Kantian philosophy has never even tried to challenge, and as such from Kantian point of view, she is addressing a problem that is not a problem.

 

The reason why I said her terminology is useless in describing this problem is that, when using her definitions for the words referring to the associated philosophical concepts, any communication about the Kantian problem (i.e. where Kant would refer to noumena) just comes out with exactly the same words as what naive realism would use to suggest an absolutely ridiculous position, that world is as we see it, despite the fact that we keep changing how we see it. Just read your post the way it reads with naive realistic definitions. See what I mean?

 

Furthermore, why did you say you disagree with my comment, when in the text you just changed the terminology? Why do you think it is important to talk about this with Rand's terminology? If anything in lost in translation, it would have to be lost to the detriment of Rand's version.

 

When you read my version, and when you wrote your response, you thought something of significance is sorely lacking in my version. This would be only because you insist on interpreting my text with your Randian terminology. And you wrote your response in Randian terminology without saying so, so either you just expected me to just happen to notice your terminology switch, or you think her terminology is somehow correct in ways that other terminologies are not? You know how DD always says you simply refuse to work under his definitions? This is exactly what that means.

 

The fact that you blatantly responded with Randian terminology is interesting because it implies exactly the obliviousness to the problem that Kant is addressing; The very fact that a terminology of any kind has been established, has already closed some real possibilities out of the mind. If you really think the only correct way to address epistemology is with Randian terminology (which is what your "completely disagreement" implies), you have already locked yourself up to only one way of thinking, and you think other perspectives are worthless; you have fallen prey to exactly the problem Kant is pointing out. From this perspective, I see Rand's philosophy as merely a single branch of an entire tree that Kant is referring to (unbeknownst to Rand herself).

 

 

The second half of your post reflects an issue that really rubs me the wrong way, which is this overwhelming desire that people have to let some chosen authority do the thinking for them. That is why the world is full of all kinds of cults and religions, filled with mindless robots who want nothing more than to have someone tell them what to believe. This sad issue is very pronounced in scientific world too.

 

I just happened to read this little article on Nature;

 

http://www.nature.com/news/stephen-hawking-there-are-no-black-holes-1.14583

 

And what's the very first sentence saying? That here's an idea that should be taken seriously because of who is saying it.

 

Why is it, that a supposedly scientific publication can imply that the validity of an idea depends on who is delivering it? What they are really alluding to is the fact that most people, including most scientists, are not interested of thinking of things for themselves. They rather choose to listen to authorative figures. It's much easier that way.

 

It wouldn't bother me as much that people don't want to think, if then those same people didn't spending their time defending ideas that they don't understand themselves.

 

This just all reflects the fact that most people really want to believe into something rather than admit they really just don't know. Even though it is the very basis of scientific thinking, that we do not know, and we cannot know. We can merely attain an understanding of sort.

 

But, even most scientists just want to believe. That is why there is always such an emotional response to anything implying a theory X has been wrong about something. Emotional response towards invalidating a theory? Give me a break. Polls asking physicists which quantum interpretation they believe in? Why believe?

 

The nice thing about DD's analysis is that it is entirely epistemological, meaning there are no expensive experiments to be performed to verify its validity. All you need is to follow logical steps, i.e. rack your own brain. No one is asking you to believe into any hypothetical thing, and no one is imagining that any of it proves the existence of any specific order of things. The fact that you still want someone else to do the thinking for you is disturbing, albeit statistically quite expected.

 

If you find his work to appear valid to you, the only thing that happens is it displays more explicitly the redundancy of assuming ontological existence to the defined objects in themselves (just like science has made the concept of Gods redundant, not dis-proven, but redundant), and more importantly it sheds light to not-so-obvious connections between some physics definitions (so unobvious, that physicists still have not found the connection between QM and GR with by the guess-work that physics normally is). That is to say, it is not a replacement of Schrödinger or Relativity in itself, but it can be seen as an analytical platform for re-arranging the terminologies meaningfully, to come up with alternative forms of the same relationships, which may offer useful ways to think about such and such real situations.

 

I haven't seen the comments of any professional physicists pointing out any errors in the analysis, but it would be nice to get clear feedback, such that it could be meaningfully addressed in the blog. So far I have pretty much just seen people confusing it for a theory of some sort, or vaguely speculating that some parts of it entail an unfair assumption, without good explanation as to why.

 

This should be pretty simple. Either the opening moves contain an unwarranted assumption, or they don't. After that, either the math checks out, or it doesn't.

 

Do you realize that even if there exists an unwarranted assumption in the opening moves, identifying that assumption merely implies that the same assumption exists somewhere in the definitions of modern physics?

 

It is possible, even likely, that the math contains some errors. And what if there is a critical error that just so happened to make Schrödinger, Dirac, and relativistic relationships to fall out unfairly? That would be a pretty amazing co-incident wouldn't it?

 

Your comments do imply you still see it as a physics theory of some sort, so perhaps it would force you to think about this issue a bit more if I asked you to explain clearly in your own words what parts of this do you see as a physics theory? Please, take your time, as no one can think things through in a hurry.

[Rade]

AnissH,

 

To answer your last question, I do not think that the analysis presented by DD is any kind of theory, he has said so repeatedly. 

 

==

 

"So it appears the critical mistake is to assume that the particles we have defined are also in themselves real objects".....AnissH==

 

I understand that your comment above is a minority view of physicists that work under a non-realist philosophy.   I do not agree with it for the reasons stated above, which I again repeat:

 

"There is absolutely no 'critical mistake' to know that particles that exist as real objects that are experienced by a human with unknown in-themselves nature have the potential to be differentiated from other such particles and thus formed into mental CONCEPTS. There is no critical mistake to then assign context dependent DEFINITIONS on these concepts to allow humans to communicate how such real particles are different from each other, and perhaps of greater value, to communicate among humans how such real particles can be mentally integrated into new concepts that allow for new definitions."

 

For me, your comment that a scientist 'cannot know' but can only attain 'an understanding of sort' is a confusing comment in light of the fact that that DD has stated often that 'the past is what you know'.  So, clearly you do not agree with DD concerning knowledge of the past, but this raises a question...what do you and DD mean by 'to know', given that you have different understandings of the concept ?   

 

If you wish to discuss Rand I suggest you begin a new thread topic, perhaps it would receive some interest by others.  Of course Rand does not agree with Kant on many issues, but who cares, most philosophers do not agree with Kant on one point or another, likewise Rand.   And, for your information, Rand does not hold the axiom that existence exists because of anything said by Kant, it fact, if you read Kant you will see that he comepletely agrees with Rand on this point. 

 

 

 

 

 

[Doctordick]

"There is absolutely no 'critical mistake' to know that Angels exist as real entities that are experienced by a human with unknown in-themselves nature have the potential to be differentiated from other such entities and thus formed into mental CONCEPTS. There is no critical mistake to then assign context dependent DEFINITIONS on these concepts to allow humans to communicate how such real Angels are different from each other, and perhaps of greater value, to communicate among humans how such real Angels can be mentally integrated into new concepts that allow for new definitions."  

 

Oh yeah, my definition of "the past" as "the past is what you know" is meant to be what your explanation is based upon.  Please correct that definition to "the past is what you think you know!"  If you think you know that Angels exist then your explanation had better be consistent with that belief.  

 

Have fun -- Dick

[AnssiH]

To answer your last question, I do not think that the analysis presented by DD is any kind of theory, he has said so repeatedly.

What would you characterize it as?

 

I understand that your comment above is a minority view of physicists that work under a non-realist philosophy.   I do not agree with it for the reasons stated above, which I again repeat:

 

"There is absolutely no 'critical mistake' to know that particles that exist as real objects that are experienced by a human with unknown in-themselves nature have the potential to be differentiated from other such particles and thus formed into mental CONCEPTS. There is no critical mistake to then assign context dependent DEFINITIONS on these concepts to allow humans to communicate how such real particles are different from each other, and perhaps of greater value, to communicate among humans how such real particles can be mentally integrated into new concepts that allow for new definitions."

Do you think that full caps emphasis somehow tells people how those terms are meant to be interpreted? Do you realize that Kant and Rand mean completely different things (and Rand seems to be oblivious about this too). I didn't say Rand disagrees with Kant. She thinks she does, but it seems more like her misunderstanding of Kant. I don't think she understands at all what Kant was saying, but she thinks she does because she had a meaningful interpretation in her head.

 

For me, your comment that a scientist 'cannot know' but can only attain 'an understanding of sort' is a confusing comment in light of the fact that that DD has stated often that 'the past is what you know'.  So, clearly you do not agree with DD concerning knowledge of the past, but this raises a question...what do you and DD mean by 'to know', given that you have different understandings of the concept ?

I think this has been pointed out to you multiple times, and I don't understand why you are not even trying to create a meaningful interpretation. Perhaps it would be easier for you if you tried to understand the steps to Schrödinger, because as always, understanding the broader context can help you in making a more correct interpretation of what he is saying.

 

Anyway, the definition of the past here is simply a reference to some "noumenaic data", which is to say, "knowing it" is not referring to knowing a correct explanation for it. What you think it is (and how do you fill the blanks), is a function of how you explain the data.

 

The only reason he is even defining such a thing as "past" here is to refer to the idea that noumenaic information keeps changing (keeps accumulating), and a definition (or definitions) for time can be created on that fact.

 

I'm sure I've suggested different words than "know" here for you multiple times but you don't really seem to be interested of trying to understand this.

 

If you wish to discuss Rand I suggest you begin a new thread topic, perhaps it would receive some interest by others.  Of course Rand does not agree with Kant on many issues, but who cares, most philosophers do not agree with Kant on one point or another, likewise Rand.   And, for your information, Rand does not hold the axiom that existence exists because of anything said by Kant, it fact, if you read Kant you will see that he comepletely agrees with Rand on this point.

I'm starting to think your interpretation of Rand may be simply naive realism, which I doubt is what she meant. But maybe she did.

 

And no I'm not really interested of talking about her philosophies too much, seems like a waste of time to be honest. The only reason for referring to it as much as I have so far is to make you understand that words have different semantics, and you can't just blatantly ignore that.